NONLINEAR ∗-LIE N-TYPE DERIVATIONS ON ∗-ALGEBRAS

Abstract

Let $\mathcal{M}$ be a $\ast$- algebra containing a non-trivial projection with unit $I.$ In this paper, we study the characterization of nonlinear $\ast$-Lie type derivations on $\ast$- algebras. For any $S,T\in\mathcal{M},$ a product $[S,T]_\ast =ST-T^\ast S $ is called $\ast$-Lie product. In this article it is shown that, if a map $ \Theta : \mathcal{M} \longrightarrow \mathcal{M}$ (not necessarily linear) satisfies $\Theta(q_{n}(S_{1},S_{2},\ldots ,S_{n})) = \sum_{i=1}^{2} q_{n} (S_{1},\ldots,S_{i-1}, \Theta (S_{i}),S_{i+1},\ldots,S_{n}) (n\geq 3)$ for all $S_{1},S_{2},\ldots ,S_{n} \in\mathcal{M},$ then $\Theta$ is additive. Moreover, if $\Theta(iI) $ is self- adjoint, then $\Theta$ is an additive $\ast $-derivation. As an application, we can also apply our result on von-Neumann algebras, standard operator algebras and prime $\ast$-algebras.